Electrolytic analogue



Feb. 25, 1958 5, o o s Y I 2,824,689 I ELECTROLYTIC ANALOGUE Filed Nov. 7, 1951 5 Sheets-Sheet l Julius S.Aron0fs1 y INVENTOR.

'ATTORNEY Feb. 25, 1958 J. s. ARONOFSKY 2,824,689

ELECTROLYTIC ANALOGUE Filed Nov. 7, 1951 S'Shets-Sheet 2 8 ":3 1 (C I .0 '0 i i xx m 5 s N Q a7 \i m N r I o \N v m g as g ATTORAWY 1958 I J. s. ARoNoFsKY 2, 4,689

ELECTROLYTIC ANALOGUE Filed Nov. '7, 195} 5 Shets-Sheet 5 INVENTOR.

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ATTORNEY Feb. 25, 1958 v J. 5. ARONOFSKY ELECTROLYTIC ANALOGUE 5 Sheets-Sheet 4 Filed Nov. 7, 1951 Julius S.Ar0n0fS'ky' I INVENTOR.

ATTORNEY v Feb. 25, 1958 .1. s. ARONOFSKY 2,

ELECTROLYTIC ANALOGUE Fil d N 7 1951 I 5 Sheets-Sheet 5 Julius siAronofsky" INVENTOR. M 6, W

ATTORNEY United States Patent ELECTROLYTIC ANALOGUE Julius S. Aronofsky, Dallas, Tex., assignor, by mesne assignments, to Socony Mobil Oil Company, Inc., a corporation of New York Application November 7, 1951, Serial No. 255,271 11 Claims. (Cl. 235-61) This invention relates to the solution of mathematical problems in systems having axial symmetry and relates more particularly to the solution of mathematical prob lems involving flow of various media in such systems by means of an electrolytic analogue. I

In many systems having axial symmetry, mathematical problems arise with respect to the flow therethrough of various media. Examples of such problems arise in connection with the logging of wells drilled into the ground, the production of oil, gas, or water from such wells, and the study of the fiow of heat through various objects such as internal combustion engine pistons or spark plugs. Essentially, the mathematical problems involve a solution to Laplaces equation which, for a cylindrical coordinate system characterized by the radial coordinate r, the vertical coordinate ,5, and the angular coordinate 0, may be written for any scalar quantity V in the form:

6 1b 1 a b V V-(y l' l' F2+555 V-0 (1) In an axially symmetrical system, there will be no variation with respect to the angular coordinate and the above equation may be simplified and written in the form:

2 23 L E) V Or rOr bE V4) (2) Solutions in analytical form have been found for the above equations in a few idealized cases. However, the solutions are dilficult and tedious to carry out even for the few idealized cases and for most practical problems it is impossible to find a solution in analytical form.

For the approximate solution in many cases to Laplaces equation, various non-analytic methods have been employed. These non-analytic methods involve the use of analogues, such as membranes, electrical networks, or analogues such as electrolytic analogues geometrically similar to the system being studied. In these analogues, a scalar quantity which is measurable obeys Laplace's equation and the behavior of the measurable, scalar quantity in the analogue is analogous to the behavior of the same or a different scalar quantity in the axially symmetrical system for which the solution to the equation is desired. Thus, by measuring one scalar quantity in the analogue and thereby arriving at a solution'to Laplaces equation for the analogue, an approximate solution is obtained with respect to the same or different scalar quantity in the system represented by the analogue. However, the use of membranes is limited to particular cases and while electrical networks can be used ina larger number of cases, a practical disadvantage resides in the extremely large number of electrical elements that must be provided for even a simple problem. With respect to the use of models geometrically similar to the system being studied, a difficulty arises in providing practical methods for representing regions within the system having different physical properties affecting flow.

It is an object of this invention to provide an electrolytic analogue. It is another object of this invention to ice provide a means for arriving at an approximate solution to Laplaces equation for systems having axial symmetry. it is another object of this IHVeIItlOHlO provide a simple means for representing in an analogue regions within a system having different physical properties affecting flow. it is a more particular object of this invention to provide means for the solution of mathematical problems involving flow effects about a well bore drilled into the ground. Further objects of the invention will become apparent from the following description thereof. i In accordance with the invention, electrical current is passed through a unitary body of a solution of electro lyte having thicknesses over a plurality of areas which vary in terms of straight line functions, each area being representative of a region within axially symmetrical system having different physical properties affecting flow and each 'straight' line function expressing the thicknesses through the various areas beingrelated by the' same factor to the physical properties affecting flow of the regions within the system, and measuring at least at one location the magnitude of the potential gradient'set up within the electrolyte by the passage of the electrical current. i '2 In the practice o f the invention I prefer to employ a trough containing a solution of an electrolyte and whose bottom surfacein contact with the solution of electrolyte consists of areas having defined linear slopes with respect to the horizontal surface of the solution of electrolyte. Each area at thebottom of the trough represents a re gion withinthe axially symmetrical system and the shape of the outline of the trough and the areas therein are geometrically similar to the outlines of a vertical plane through the system and the regions thereof that they represent over'at least that portion of the system where the fiow of the medium occurs or the physical properties of the regions affect the flow of the medium, and the thicknesses, in this case the depths, of the solution of electrolyte over these areas,ras governed by the slopes of the bottom of the trough at these areas, are related to the physical properties affecting flow of the regions within the axially symmetrical system. By providing a body of a solution of electrolyte having thicknesses representing regions in an axially symmetrical system having different physical properties affecting flow, the invention is applicable to a large number of cases, the requirement for a large number of electrical elements is eliminated, and the various regions of the system are easily represented. The invention is applicable to the solution of mathematical problems involving flow of any medium in a system having axial symmetry. A system having axial syminetry may be defined as one whose physical properties or characteristics are expressible in terms of values of a radial coordinate, a vertical coordinate, and an angular coordinate, but which has no variation in at least one physical property or characteristic along one of the coordinates. An example of such a system may be ,a well 'drilled into the earth. The fiow of any medium such as electrical current, gas, water, oil, etc., .in this system is expressible by Laplaces equation and, accordingly, the invention is applicable to the study of and solution to .problems in the loggingof-wells by electrical methods and the production of petroleum oil or gas from ,a subterranean reservoir. Examples of such other systems are pistons or spark plugs in internal combustion engines and the flow of heat in these systems is expressible by Laplaces equation. Thus, the invention is also applicable to the study and solution to problems in the design of pistons, spark'plugs, and other apparatus.

The invention will now be described in greater detail in connection with the accompanying drawings which illustrate various specific embodiments thereof.

Figure 1 is a diagrammatic setcional view of a well bore penetrating various earth stata and illustrates electric logging of the well.

Figure 2 is a perspective view, partly in section of an analogue constructed in accordance with my invention for study of the fiow of electriclogging currentin .the system represented in Figure 1.

Figure 3 is a plan view of a subterranean oil reservoir having input wells and an output well.

Figure 4 is a diagrammatic sectional view of a portion of the subterranean oil reservoir of Figure 3 containing one input well and theoutput well and illustrates secondary recovery of the oil contained. in the reservoir by water flooding.

Figure 5 is a perspective view,'partl-y insection, of an analogue constructed in :accordance with my inven tion for study of the fiow of fluid in .the .system illustrated in Figure 4.

Figure 6 is a cliagrammaticisectional view of a cylinder and piston in an internal combustiontengine.

Figure 7 is a perspective view, partly in ,section,.of an analogue constructed in accordance with my invention for study ofthe flow of heat in the piston illustrated in Figure 6.

Referring now to Figure 1, a well 10 in the. ground 11 penetrates various strata such as stratum 12, stratum 13 stratum 14, stratum 15, and stratum 16. extending vertically fromthe axis of thewell, and'is filled with drilling fluid 17. It will be assumed that strata 12. 14, and 16 are of the same type of earth material and have the same physical properties afiecting the flow of electrical current, i. e., the .same electrical conductance. Strata 13 and are two different types .of-earth material andare of different earth material than strata 12, 14, and 16. Thus, strata 13 will have different physical properties afiecting the .fiow of electrical current than strata 15 and each of these strata will have different physical properties affecting the flow of electrical current than strata 12, 14, and .16. Additionally, the drilling fluid 17 will have different physical properties affecting the flow of electrical current than those of any of the strata. In this system, the value of any particular physical property, such as the electrical conductance will vary along the vertical coordinate since the value of the physical property varies witheach stratum. Since the portion of the system extends from the center of the well, the vertical coordinate will be regarded .as running vertically at the center of the well. While the specific electrical conductivities of the various strata and the drilling fluid maybe assumed to be uniform withhorizontal distance from the well, the value of the electrical conductance measured from the well to anypoint in question will vary with the distance from the center of the well. Thus, values of any physical property of the strata will vary along the radial coordinate, i. e., a line running horizontally from the center of the well. The values .Of any physical property will be the same at anyone distance from the well irrespective of the orientation about the well and,

therefore, values of any physical property will not'vary along the angular coordinate, i. e., a line running in a circle about the well and having its "axis at the center thereof. This system, therefore, has complete symmetry with respect to the vertical axis.

Assuming that it is desired to obtain an electrical resistivity log of well 10, a sonde20 is lowered 'down the well by means of multiconductor cable'21 leading to unit 22 containing a source of power and arecorder. Electrical current is sent through cable 21 to current.electrodes 23 and 24 and the current distributes itself through the various strata flowing from one .currentlelectrode 'to the other. The flow of the electrical current from one current electrode to the other through the various strata and the drilling fluid sets up :a potential field therein whose values are affected :by the electrical conductance of the various strata and the drilling'fiuid "and by measurement of the potential field at various points in the well valuable information concerning the strata penetrated by the well is obtained. The values of the potential field are measured by means of potential electrodes 25 and 26 connected through cable 21 with the recorder in the unit 22, and a record of the variations in values of the potential field with depth is obtained. However, the record requires interpretation to translate the variations in the potential field into a. physical picture of the strata and considerable uncertainity and differences of opinion enter into interpretation of the log. By solving Laplaces equation for the flow of electrical current through various well systems, a great deal of information which would assist in interpretation of the log can be acquired, such as the effect of different configuration of strata, the effect of different physical properties of the strata, or the eifect of variation in the electrical current employed for the logging.

With the apparatus of Figure. 2,.a solution to Laplaces equation for the flow of electrical logging current in the system illustrated in Figure 1 may be obtained. Referring to Figure 2, the analogue comprises a trough 30 containing an electrolyte solution 31 and having a bottom 32 which slopes with respect to the upper horizontal surface of the electrolyte solution. .Resting on the bottom 32 of the trough are two wedge shaped members 33 and 34 and extending .along one wall of the trough is asloping channel 35. Positioned within the channel 35 is a sonde 40 provided with two current electrodes 41 and 42 and two'potential electrodes 43 and 44, and the sonde is slidably movable in the longitudinal direction of the channel. Current electrodes 41 and 42 are connected by cables 45 and 50, respectively, with an alternating currentpowersupply 51.and potential electrodes 43 and 44 are connected by cables .52 and 53, respectively, with voltage recorder 54.

It will be seen that trough 31 is divided into a plurality of areas, namely, :area60 defined by the channel 35, area 61 between the wall62 of the trough and member 33, area 63 .defined by member 33, area 64 between members 33 and 34, area 65 defined by member 34, and area 76 between member 34 and wall 71 of the trough. Each of these areas is representative of a region within the axially symmetrical system illustrated in Figure 1, namely, area 60-is representative of well 10 from the axis .to the side walls, area 61 is representative of stratum 16, area 63 is representative of stratum 15, area 64.is representative of stratum 14, area 65 is representative of stratum13, and-area is representative of stratum 12. Line 72 representsthe-vertical axis of well 10 and eacharea slopes from, or-theprojection of the surface of the area slopes from, thislinc. The slopes of areas 61, 64, and 70are.thesame,-the slope of area 60 differs from the slopes of areas-61, 63, and 70, the slope of area63ditfers fromthe slope of area 60, and the slope of area 65 differs from the slope of area63.

In the trough of Figure 2, thepotentials set up by the flow of electrical current through the electrolyte 31 betweenelectrodesll and 42 can be expressed by the equation a by '6 6v where x is the distance along one. horizontal coordinate, y is the distance. along the other horizontal coordinate, v is the voltage, and .8 is the equivalent electrical conductivity of the electrolyte, the equivalent conductivity of the electrolyte being the product of the specific conductivity and thedepth of the electrolyte.

Equation 2 previously mentioned may be written in theform:

where r and E; are independent variables and Equation 4 reduces to 5? br) os oz) (5) which equation is analogous to Equation 3 for the trough. However, the effective conductivity 8 of the electrolyte must be made proportional to the radius r for complete analogy between Equations 3 and 5 and this is accomplished in accordance with my invention by making the thickness, or depth, of the electrolyte proportional to the radius r, or radial coordinate.

As stated previously, the value of the radial coordinate of the system illustrated in Figure 1 varies with distance from the center of the well 10 and the variation is effected in the trough by providing a constantly increasing thickness or depth for each area representing regions in the system having axial symmetry, i. e., by providing a sloping bottom to each area. Additionally, the thicknesses of each area must vary from each other since they are made proportional to the radial coordinate r whose value for each region in the system possessing axial symmetry is dependent upon the physical property of the region governing the flow therethrough of the particular medium of interest and these physical properties vary between themselves. This is effected in the trough by increasing the thickness of each area at a different rate, i. e., by providing each area with a slope proportional to the physical property of the region represented by the area. The slope of each area is expressible mathematically as a straight line function given by the equation where m is the slope of the area and 0 is the angle between the bottom and the surface of the electrolyte solution. Further, the slopes of each area will bear the same relationship to each other as the physical properties affecting flow of each area bear to each other. Thus, if the value of the physical property favorably affecting flow in one region is twice the value of the physical property favorably affecting flow of another region, the slope of the area representing the second region will be onehalf the slope of the area representing the first region.

m=tan 0 for each area. Thus, the thickness of each area is expressible in terms of the slopes, m, of the area, a straight line function, and the slopes of the various areas are related by the same factor, k, to the physical property, in this case, p, of the region within the system to be studied.

The particular physical property of the regions of the system illustrated in Figure 1 affecting the flow of the electrical logging current therethrough is electrical conductivity and therefore the slopes of the areas of trough 30 will vary in proportion to the electrical conductivity of the regions they respectively represent. In order to determine the slopes of the areas of the trough, itis necessary to know the specific electrical conductivities of the drilling fluid 17, the electrical conductivities of strata 12, 14, and 16, and the electrical conductivities of the strata 13 and 15. The specific conductivity of the drilling fluid may be determined by direct measurement on a representative sample of the fluid and the specific conductivities of the various strata may be obtained by direct measurement on samples of each stratum obtained by side wall coring or on samples of each stratum obtained at the time the well was drilled. Knowing the various specific electricalconductivities of each region, slopes for the" 'where p is the value of the physical property of the re- :gion affecting flow and k is a constant which is the same seamen Expressed mathematically, the slope of each area will be given by the equation areas in the tr'ough representing the regions may then be determined employing Equation 7, selecting any arbitrary value of k. The value of k, ordinarily, should be selected such that the slopes of the areas will not be so vastly different that the areas may not be conveniently provided within a trough.

As mentioned before, the shape of the outline of the trough and the areas therein are geometrically similar to the outlines of a vertical plane through the system and the regions thereof that they represent over at least the portion of the system where the flow of the medium occurs or the physical properties of the regions affect the flow of the medium. The flow of the electrical logging current extends for a distance into the strata from the well, and the geometric shape of a vertical plane through the regions of the system through which the flow of the electrical current extends or is affected is unknown since this is one of the matters determinable by solution of Laplaces equation for flow of the electrical current in the system. However, since the shape of the outline of the trough, and the areas therein, need be geometrically similar to the outline of a plane extending over only at least the portion of the system where the flow of the electrical current extends or is affected, a greater portion of the system may be selected for the representation in the trough. Thus,

an area of the system extending outwardly from the well to a distance beyond which the flow of the electrical current could not possibly occur or be affected may be selected for representation and the shape of the outline of a vertical plane passing through this section can be rectangular. The shape of the trough will accordingly be rectangular. The shapes of the outlines of the areas within the trough representing-the strata and the well will be geometrically similar to the shapes of the'outlines of a vertical plane extending through the strata and the well and the shape of the outlines of the vertical plane passing through the strata at the distance beyond which the ele'ctrical current could not flow or be affected will correspond to the rectangular shape of the outline of the vertical plane of the area of the system selected for representation. In the system illustrated in Figure 1, the strata extend horizontally and parallel from the well and hence the areas representing the strata will be rectangular. Should it be known that the thicknesses of the strata in the systern vary with distance from the well or that the strata curve with respect to a perpendicular to the axis of the well, the areas in the trough will similarly vary in width or curvature. The shape of the outline of the well is similarly rectangular and thus the area inthe trough representing the Well will also be rectangular.

The physical dimensions of each area within the trough are then made on the basis of a scale relationship with the physical dimensions of the regions represented by the areas. The diameter d of the well 10 as well as the Width and the distance separating the electrodes 23, 24, 25, andthe dimensions for the areas in the trough. Thus, assuming the scale factor k to be 0.01 and the thickness of strata 13 to be 50 feet, the horizontal width w, of area 65 representing strata 13 will be 0.5 foot-or'6 inches. Similarly, if the diameter of the well is 6 inches, the Width w of area 60 representing the well will be .03'inch, i. e., one-half of .06 inch since area 60 represents well 10 only from its axis to the side walls. The same factor is applied to the thicknesses of strata Hand 15 to obtain the widths of areas 64 and 63. With'respect to thewidths w of areas- 61 and 70 representing strata 16 and 12, respectively, as-

suming the thicknesses of these strata to be several times the length of the sonde 20, these widths need not be accurately scaled since theelfect of increasing thicknesis'ofj these strata onthe flow of electrical current in strata 13,

14,'and-15-becomes negligible afterthe thickness of these strata exceeds by about ;two or three times the lengthf sonde 23: The length of the areas, i. e., the distan c'it'he" area extends perpendicular to the WldllTW, should be such that extension of the length would not affect thefi ow of the electrical current in the trough} again assuming that the length' of the regions they represent is suchthat a change in the length'would not affect the flow of theelectrical current therethrough'. There are two meansfor ef-" stantthat Wlllj'glVC the desiredcondition for the area rep:

resenting the region of greatest conductivity.

The same considerations are applied with respect to selecting dimensions for the sonde 40.- The relationship of the size and spacing o f the current electrodes and the potentialelectrodcs in the sonde 40 should be the same as the relationship of; the size andspacing of the current electrodes and potential electrodes in the sonde 20. Thus, the same scale factor k is applied to the dimensions of the various electrodes andthe distance between the various electrodes for the sonde 40;

Having selected dimensions for the sonde and dimensions and slopesfor the areas in the trough, the sonde and the trough are constructed-in accordance therewith. It has been found convenient to construct a trough having walls 62, 73; 71 and 74 and provide therein a bottom covering the entire distance-between the walls except for the area 60, the bottomhaving the slope required for the area-of-greatest width and-the projection of the upper surface of the area leading to line 72-. In this way, the other areas having the selected dimensions and slopes may be provided simply by positioning andlfastening wedge shaped members having the selected dimensions and slopes, such as members 33gand34, on the bottom 32, the projection of the upper surfaces likewise leading to line 72. Another advantage resides in this arrangement, namely, the trough with the four walls and the bottom 32 may be employedfor the study of systems other than the system illustrated in Figure) by selecting a slope constant k, Equation 7; such that the slope of bottom 32 will be the desired slope for any region of the system and positioning wedge shapedmembcrs of the proper dimensions andslopes on the bottom. However, if desired, each area may comprise an individual, sloping member positioned and fastened within the four wallsof the trough or all or a number of the areas may be formed from a single memher, as by shaping the single member to provide all or a number of the areas.

The area 60 may be provided by positioning andfastening a member 75 against wall 74 and then positioning and fastening bottom 32 against member 75. However, if desired, bottom 32-may extendto wall 74 and channel 35 providing area 60 of the proper slope formed therein. Sonde 40 is shaped at-its bottom portion to conform to the area 60 and is maintained in position in channel 35 by means of dovetailed track 76 and a tongue on the sonde fitting therein. Further, sonde 40 extendswithinarea 60 from wall-74 theproportional distance that sonde 20 extends beyond-the axis of well 10.

The trough 3t) and the other elements therein except the electrodes 41, 42, 43, and 44are made of any suitablematerialsuch as wood, plastic, etc. Preferably, the material; employed should be electrically non-conducting and impervious to aqueous electrolyte solutions but, if not, suitable coating materials such. as paraffin, resin, varnish, etc., may be applied to the surfaces thereof exposed to the electrolyte solution. to prevent flow of electrical current througlrthe elements and prevent absolution of electrolyte between the elements which would vary the effective depth of the solution 'of electrolyte The solution of electrolyte employed may be an aqueous solution of any desired electrolyte such as sulfuric acid,

sodium chloride, copper sulfate, etc The eleetrolyte.

employed may be a salt of the same metal of which theelectrodes 41,42 43,:andA4, are. composed but other electrolytes may be employed; The concentration of electrolyte in the solution maybeas desired, the only limitatiqn iherie I being imposed ,by the necessity of obtaining currents r i v and voltage recorder 54 with anygiven practical potential appliedacross cur'rent electrodes 41 and 4 2. The amount offthejsolution of electrolyteemployed must be s uflicient to just fill the trough to the levelof line 72 representing the axis of thew ell.

For solution of Laplaces equation for the flow of. the electrical logging current, in the Iwell system illustrated inL Figure 1, a 'suita ble ,alternating current from power: supplySljs appliedv to current .electrodes 41 and 42 and thesonde isrnoved along the .length of wall 74. As the.

illustratedin Figurell arehighly desirable from several aspects. Assumingthat thewell systemis believed to be as illustrated in Figure l and hthe trough. is set up as illustrated ,in Fjgur e 2 but thenelectric al resistivity log of the well is not in accordance with the solution to Laplaces equation obtained by useoil the analogue, it may then be assumed thatjhe well system is no t as pictured. By providing the analogue with areas ,having different di;

mensions and slopes, soliitions to" Laplaces equation may" be obtained until solution obtained which agrees with the electrical resistivity log of the well Having obtained this solution a more accurate interpretation o'f the: log,

of the wcllis obtained Further, assuming that it is desired to determine the effeetof the conductivity of the drilling fiuid 17011 the log of the well, the slope of area 60 may be varie d and solutions to Laplaces equation obt ained; for each slope, thereby. also making possible a more accurate interpretation of the log of the well. In similar manner, the effect of variation in electrical conductivity, width, etc., ofthe various strata on the log of the well may be determined.

It was mentionedpreviously that the specific conductivities. of thewarious strata may. be obtained by direct measurement onsamplesiof each stratum obtained by side wallporing or.on samples of cachstrat um obtained at the time the well.wasdrilledand -that, knowing the specific electrical conductivities ofithe strata, slopes for the areas inthe trough representing these stratamay be determined employing Equation 7, selecting. any arbitrary value of k. Inmany instances, however, adependable measurement ofthe specific conductivities of the various strata cannot be obtained beeause of the change of fluid content in the samples of the strata upon removal of the samples from thestrata and bringing them to the surface of the earth. Accordingly, a particularlyvaluable use of the analogue resides in determination of the specific conductivities of the strata by varying the slopes of the areas representing the strata until asolntion to Laplaces equation is in agreement with the electrical resistivity log of the well. Knowing the slope of the areas, the specific conductivities 1 fig 'may be;determined mp q t n h myemrioecranralsg be employed for the solution of problemsin the production oflpetroleum oil or gas from subterranean petroleum reservoirs. Anexample of.

12 a. wh n rises, s. eastern m v r.

surablqby potential, electrodes 43 and 44 petroleum oil from a subterranean reservoir by means of a gas drive or water drive. In this secondary method of recovery, water or gas is forced into a petroleum reservoir through an input well or wells leading thereto and the water or gas drives the oil through the reservoir to an output well leading therefrom through which the oil may be recovered. Figure 3 illustrates a typical seven spot well spacing employed for secondary recovery of oil from a subterranean petroleum oil reservoir by water drive. Reservoir 101 is penetrated by six input Wells 102, 103, 104, 105, 110, and 111 and by an output well 112. Water is forced into the input wells and drives the oil contained in the reservoir toward the output well from which it may be recovered. It will be assumed that input wells, 102, 103, 104, 105, 110, and 111 are uncased and that output well 112 is cased.

' Referring to Figure 4, input well 105, which is similar to the other input wells, is provided with well head 113 containing inlet pipe 114. Well 105 penetrates strata 115 and 120 overlying stratum 101 which is the oil reservoir and it will be assumed that strata 115 and 120 are permeable to the flow of water therethrough, stratum 120 being less permeable than stratum 101 and stratum 115 being less permeable than stratum 120. Output well 112 is provided with well head 121 containing outlet pipe 122 and is also provided with casing 123 which extends from the well head through strata 115 and 120 and for a distance into stratum 101. By virtue of the facts that well 105 is uncased, well 112 is cased through strata 115 and 120, and strata 115 and 120 are permeable to the flow of water, the water pumped through well 105 will flow through strata 115 and 120 as well as stratum 101 but will have to flow downwardly from strata 115 and 120 into stratum 101 to enter output well 112. The system illustrated in Figure 4 approximates axial symmetry about well 112, the strata extending vertically from the axis of the well, and the flow of the water in the system will obey Laplaces equation.

Solutions to Laplaces equation for the flow of water inthe system of Figure 4 can be obtained with the electrolytic analogue of Figure 5. Referring to Figure 5, the analogue comprises a trough 124 having walls 125, 130, 131, and 132 and bottom 133 divided into areas 134, 135, and 140. Extending along the entire length of the wall 1 32.is.a11 electrode 141 which adjoins the areas 134, 135,

and 140 and extending along a portion of wall 130 is an.

electrode 142 which adjoins a portion of the area 134. The trough contains electrolyte solution 143 and con nected between electrodes 141 and 142 is an alternating current power supply 144. Area 134 represents stratum 101, area 135 represents stratum 120, and area 140 represents stratum 115. The projection of each area extends topa line, not shown, along the inner surface of wall 130, which line is taken as the axis of well 105. An electrically non-conducting member 145 is positioned along the portion of wall 130 not occupied by electrode 142. In this analogue, the flow of electrical current between the electrodes 141 and 142 is analogous to the flow f the water from well 105 to well 112.

Permeability is the physical characteristic of strata 101,

120, and 115 governing the flow of water therethrough.

and the slopes of the areas 134, 135, and 140 with respect to the surface of the electrolyte solution will be proportional to the permeability of the strata they represent. The slopes will be given by Equation 7 where p will be the value of the permeability of each stratum and the value of k will be arbitrarily selected. The permeability of each stratum can be determined by measurements made on core samples obtained by side wall coring or obtained during drilling of the wells.

' The shape of the outline of each area within the trough is'determined similarly as the shapes of the areas within the" trough of Figure 2, and the physical dimensions of each area Within the trough are then selected, the selection being similarly made on the basis of a scale relationr 16 ship with the physical dimensions of the strata represented by the areas. The thicknesses t of the strata 101, 120, and 115 at the well 112 are determined, as, for example, from a log of the well. The distance 1 between the two wells and 112 representing the length of the strata is also determined. A convenient scale factor k is arbitrarily selected and the scale factor is applied to the thicknesses and length of the strata to obtain the dimensions of the areas in the trough. Should it be known that the thicknesses of the strata 101, 120, and 115 vary between the wells 105 and 112 or should the strata curve with respect to a perpendicular to the axis of the well 112, the areas in the trough should similarly vary in width and curvature. Having selected dimensions for the areas in the trough, the trough is constructed in accordance therewith.

Water can flow into strata 101, 120, and 115 from well 105 but can flow into well 112 only from the uncased portion in strata 101. Thus, in the analogue where the flow of electrical current between electrodes 141 and 142 will be analogous to the flow of the water from well 105 to well 112, electrode 141 extends along the entire length of areas 134, 135, and 140 but electrode 142 extends along only a portion of area 134, the distance the electrode extends along area 134 being proportional in accordance with the selected scale factor k to the length of the uncased portion of Well 112 in strata 101. Similarly, since the rate at which water can flow into strata 101, 120, and 115 will depend upon the circumference, and thus the radius, of well 105, the width w of electrode 141 must be scaled to the radius of well 105. The width of member 145 is the same as the width of electrode 142 in order that the boundary of the solution of electrolyte will extend only to line 146 representing the wall of the well. The rate of flow of water into well 112 depends upon the circumference thereof andrelectrode 142 is also scaled to the dimensions of well 112. The electrodes 141 and 142 must extend vertically from at least the level of the solution of electrolyte to, the entire depth thereof at each area.

The same types of solutions of electrolyte may be employed as in the analogue of Figure 2. Similarly, the same concentrations of electrolyte may be employed. With respect to the amount of the solution of electrolyte in the trough, the solution must fili the trough to the level of the line previously mentioned extending along the inner surface of wall 130, this line being taken as the axis of well 112. Withrespect to Figure 5, this line will be located along wall at the surface of the wall contacting electrode'142 and member 145 and will be the projection of the surface of the solution of electrolyte to wall 130.

The flow of electrical current from electrode 141 to electrode 142 through areas 134, 135, and will be in accordance with Laplaces equation for the flow of water from well 105 to well 112 through strata 101, 120, and 115 and by measurement of the potential fields set up in the solution of electrolyte by the passage of the current therethrough, solutions to Laplaces equation for the flow of the water between wells 105 and 112 are obtained. For example, with the analogue of Figure 5, solutions to problems in pressure distribution, direction of streamlines, and distribution of the water from well 105 into each of the permeable strata between wells 105 and 112 can be obtained. Pressure distribution in the permeable strata between wells 105 and112 can be determined by imposing an alternating potential on electrodes-141 and 142 from power supply 144 and measuring the potential field set up in the electrolyte solution at sufiicient points to obtain a plot of the potential field. The values of the potential field as measured will be equivalent to the values of the pressure distribution. The value of the alternating'potential imposed on electrodes 141 and 142 can be as desired as long as it is sulficiently high to obtain measurable values of the potential field set up by the flow of current device may be employed, the probes being inserted-below the level of the solution of electrolyte at locations corresponding to locationsin the permeable strata where pressure distribution information is desired. The effect on the pressure distribution of change of pressure of the water pumped into well 105 may be determined by changing the value of the alternating potential imposed on electrodes 141 and 142 in proportion to the change in the. water pressure and measuring again the values of the potential field.

For determination of the direction of the streamline of fiow of water in the permeable strata at any one point, the two electrode probes are moved about a circle in the solution of electrolyte, the center of the circle being the point in the analogue equivalent to the point in the permeable strata, until the potential measuring device reads zero, indicating that the two probes are on an equipotential line. The direction of the streamline will then be the perpendicular at the point in question to the line joining the two electrode probes.

The distribution of the water from well 105 into each of strata 101, 120, and 115 may be determined employing conventional mathematical calculations from a plot of the pressure distribution and the streamlines.

The flow of heat in a piston in an internal combustion engine follows Laplaces equation and solutions to the equation for this system can be obtained by the invention. Referring to Figure 6, piston 200 is positioned within cylinder 201 in motor block 202 and is provided with piston rings 203 and 204. The motor block contains channel 205 for cooling water and the cylinder is covered with head 210 providing firing chamber 211. The head contains intake valve 212, outlet valve 213, and spark plug 214. In operation of the engine, heat is generated within the firing chamber 211 by the burning of the fuel charges therein, and the upper surface of the piston will be exposed to this heat. The heat will flow through the piston and will leave the piston through the piston rings 203 and 204 which contact the side walls of the block 202. Some heat will leave the piston through the space between the side walls of the piston adjoining the side walls of the block and from the lower and inner surfaces of the piston and the wrist pin and connecting rod assembly 215 but this passage of heat will be negligible compared to the heat passage through the piston rings andmay be neglected. Since the heat will flow from the upper surface of the piston to the piston rings, the heat flow in the piston will depend upon the temperature differential between the upper surface of the piston and the side walls of the block in contact with the piston rings and' upon the thermal conductivity of the piston and the piston rings. The metal constituting the piston willhave a higher thermal conductivity than the metal constituting the piston rings and therefore the heat flowing from the upper surface of the piston to the side walls of the block will pass through regions of different thermal conductivity. Solutions to Laplaces equation for the flow of heat in the piston of Figure 6 may be obtained with the analogue of Figure 7 wherein the flow of electrical current will be representative of the flow of heat.

Referring to Figure 7, the analogue comprises a trough 220 whose outline is shaped similarly to the outline of a vertical section taken along the piston bisected at its axis, a suitable scale relationship being selected for the dimensions of the trough compared to the dimensions of the piston. Wall 221 of the trough represents half of the upper surface 222 of the piston from the axis thereof to the side wall, wall 223 represents the side wall 224 of the piston, wall 225 represents the bottom surface 230 of the piston, and wall 231 represents the inner surface 232 of the piston. T he bottom of the trough comprises three areas, namely area 23 3 representing the piston borly, area 234 representing piston ring 203, and area 235 representing piston ring 204. Areas 234 and 235 are formed of blocks of any suitable material and the length and scale relationship employed for the trough. Further,

the areas are positioned with respect to the trough as are the; piston rings positioned with respect to the piston body. Since heat enters along the upper surface of the piston and leaves through the piston rings, the electrical current should similarly pass through the analogue. Accordingly, electrode 241 is positioned along the entire length of wall 221 and electrodes 242 and 243 are positionedalong the lengths of areas 234 and 235, respectively, electrode 241 and electrodes 242 and 243 being connected to alternating current power supply 244. The trough is filled with solution of electrolyte 245, the level of the electrolyte solution being no higher than line 246 representing the axis of the piston to which the upper surface of area 233 extends.

Thermal conductivity being the physical characteristic of the piston and piston rings affecting flow of heat there-- through, the slopes of each of the areas 233, 234, and 235 are selected in accordance with the thermal conduc tivities of the respective elements they represent, Equation 7' giving the slope where p will be the thermal conductivity of each region and k will be arbitrarily selected. With respect to areas 234 and 235, the projection of their sloping upper surfaces extends to line 246.

The flow of electrical current from electrode 241 to electrodes 242 and 243 will be in accordance with Ilaplaces equation for the flow of heat from the upper surface through the piston rings. By measurement of the potential field set up in the electrolyte solution by the passage of'the current therethrough, solutions to Laplaces equation for the flow of heat through the piston are obtained. With the analogue of Figure 7, solutions to problems in temperature distribution, direction of heat flow, and magnitude of heat flow may be obtained. Temperature distribution in the piston can be determined by imposing an alternating current between electrode 241 and electrodes 242 and 243 from power supply 244 and measuring with a probe electrode the values of the potential field set up in the electrolyte solution, the values of the potential field being equivalent to the values of the temperature distribution. By employing a pair of closely spaced probe electrodes rotatable about a point located between the electrodes, the direction of isopotential lines.

may be conveniently determined. Direction of heat flow in the piston at any particular point will be the perpendicular to the equipotential line at the corresponding point in the trough and distribution of heat flow may be determined by employing conventional mathematical calculations.

By the use of analogues similar to the analogue of Figure 7, investigations as to the best design of piston and piston rings from the standpoint of heat transfer can be made. Various analogues can be made having different dimensions, configurations, and slopes representing different dimensions, configurations, and thermal conductivities of piston and piston rings and temperature distribution, etc., determined in the various analogues until the best design is found. Various other uses of the analogue will readily suggestithmselves.

The analogues hereinabove described are three dimensional, i. e., they have length, width, and depth. Theoretical, considerations require that the analogues be two dimensional, i. e., have length and width only, in order that the direction of flow of electrical. current therethrough not be distorted by the depth configurations of the analogues, which, distortion would not occur in the direction of flow of -the media in the systems represented by the analogues, Ordinarily, however, the depth configurations of the analogues distort the direction of flow. of the elec-' trical' current team extent that is negligible with respect to accuracy of, the solutions to Laplacefs equation obtained'by use of the analogues. In the cases where depth configurations are extreme and inaccuracies are thereby introduced into the solutions to Laplaces equation obtained by use of the analogues, the effects of the depth configurations may be eliminated by attaching thin electrical conductors at uniformly spaced points to the edges of the areas defining the regions in the system of different physical characteristics and extending the conductors to at least the surface of the solution of electrolyte whereby the electrical current will flow through the analogues as though the analogues were more truly two dimensional.

It will be apparent to those'skilled in the art that various modifications of the invention may be made without departing from the spirit thereof. For example, in place of varying the thickness or depth of the solution of electrolyte over the various areas by shaping the bottom of the body of solution of electrolyte, the depth or thickness may be varied by providing a cover to the trough, or a floating member in the solution, whose bottom surface in contact with the solution of electrolyte would be shaped to provide areas sloping with respect to the bottom of the solution of electrolyte and thereby shape the surface of the body of solution of electrolyte. Further, the thickness or the depth of the solution of electrolyte may be varied by a combination of shaping the bottom of the body of solution of electrolyte and providing a cover or a floating member to shape the surface of the body of solution of electrolyte. Also, while the outline of the vertical planes through the systems of Figures 1, 4, and 6 are represented in the analogues of Figures 2, 5, and 7 as horizontal planes therethrough, they may be represented as vertical planes therethrough by proper construction of the trough or container for the solution of electrolyte. For example, the analogues of" Figures 2, 5, and 7 may be provided with covers or other liquid containing means at the level of the solution of electrolyte and the analogues turned 90 degrees on end. In these cases, the section or wall of the analogue which would be uppermost may be eliminated if it is completely horizontal.

Having thus described my invention, it will be understood that such description has been given by way of illustration and example and not by way of limitation, reference for the latter purpose being had to the accompanying claims.

I claim:

1. In an analogue representing a system having axial symmetry and containing regions having different values of a physical property affecting flow of a medium therethrough, the combination which comprises a container for a body of solution of electrolyte, a plurality of plane surfaces constituting at least one inner surface of said container, the slope of at least one of said plane surfaces differing from the slope of one other of said plane surfaces and the projection of all of said plane surfaces leading towards a common line, and a pair of electrodes positioned interiorly of said container.

2. In an analogue representing a system having axial symmetry and containing regions having different values of a physical property affecting flow of a medium therethrough, the combination which comprises a trough having side walls and a bottom, said trough being adapted to contain a body of solution of electrolyte, a plurality of plane surfaces constituting the inner surface of said bottom of said trough, the slope of at least one of said plane surfaces differing from the slope of one other of said plane surfaces and the projection of all of said plane surfaces leading towards a common line, and a pair of electrodes positioned interiorly of said trough.

3. In an analogue representing a subterranean system having axial symmetry and containing regions having different values of a physical property affecting flow of a medium therethrough, the combination which comprises a trough having side walls and a bottom, said bottom comprising a plurality of plane sloping surfaces, at least two of which plane surfaces extend at a finite angle with of a medium in said subterranean system and subterr anea-n strata having different values of a physical property affecting the flow of said medium in said subterranean system, the combination which comprises a a trough having side walls and a bottom, said bottom comprising a plurality of plane surfaces, one of which plane surfaces extends longitudinally of said trough and one of which plane surfaces extends at a finite angle with respect to said first mentioned plane surface and at least two of which plane surfaces have slopes which differ with respect to each other, the projections of all of said plane surfaces leading towards a common line, and a pair of electrodes positioned interiorly of said trough.

5. In an analogue representing a subterranean system having axial symmetry and comprising two wells and subterranean strata having different values of a physical property affecting the flow of fluid in said subterranean system, the combination which comprises a trough having side walls and a bottom, said bottom comprising a plurality of plane surfaces at least two of which plane surfaces have slopes which differ with respect to each other, the projections of all of which plane surfaces lead towards a common line, an electrode positioned interiorly of said trough at one side thereof and extending substantially parallel to said common line, and another electrode positioned interiorly of said trough at an opposite side thereof and extending substantially parallel to said first mentioned electrode.

6. In an analogue representing a subterranean system having axial symmetry and containing regions having different values of a physical property affecting flow of a medium therethrough, the combination which comprises a trough having side walls and a bottom, said bottom comprising a plurality of plane sloping surfaces, at least two of which plane surfaces have slopes which differ with respect to each other, and the projections of all of said plane surfaces leading towards a common line within said trough, a unitary body of solution of electrolyte filling said trough to the level of said common line, and at least a pair of electrodes positioned interiorly of said trough and contacting said unitary body of solution of electrolyte.

7. In an analogue representing a subterranean system having axial symmetry and comprising a well containing a material having a physical property affecting flow of a medium in said subterranean system and subterranean strata having different values of a physical property affecting the flow of said medium in said subterranean system, the combination which comprises a trough having side walls and a bottom, said bottom comprising a plurality of plane surfaces, one of which plane surfaces extends longitudinally of said trough and one of which plane surfaces extends at a finite angle with respect to said first mentioned plane surface, at least two of which plane surfaces have slopes which differ with respect to each other, and the projections of all of said plane surfaces leading towards a common line within said trough, a unitary body of solution of electrolyte filling said trough to the level of said common line, and at least a pair of electrodes positioned interiorly of said trough and contacting said unitary body of solution of electrolyte.

8. In an analogue representing a subterranean system having axial symmetry and comprising two wells and subterranean strata having different values of a physical property affecting the flow of fluid in said subterranean system, the combination which comprises a trough hav,

ing side walls and a bottom, said bottom comprising a plurality of plane surfaces at least two of which plane surfaces have slopes which differ with respect toeach other and the projections of all of which plane surfaces lead towards a common line within said trough, a unitary body of solution of electrolyte filling said trough to the level of said common line, an electrode positioned interiorly of said trough at one side thereof and contacting said unitary body of solution of electrolyte, and another electrode positioned interiorly of said trough at an opposite side thereof and contacting said unitary body of solution of electrolyte.

9. In an analogue representing a system having axial symmetry and containing regions having different values of a physical property affecting flow of a medium therethrough, the combination which comprises a container for a body of solution of electrolyte, a plurality of plane surfaces constituting at least one inner surface of said container, the slope of at least one of said plane surfaces differing from theislope of one otherof said, plane surfaces and the projection of all of said plane surfaces leading towards a common line, a unitary body of solution of electrolyte submerging said plane surfaces to. the level of said common line, and a pair of electrodes positioned intenorly of said container.

10. In an analogue representing a subterranean, system having axial symmetry and' comprising a well containing a material having a physical property affecting fiowiofa medium in said subterranean system and subterranean strata having different values of a physical property affecting the flow of said medium in said subterraneansystem, the combination which comprises a trough, having side walls and a bottom, said bottom comprising a plurality of plane surfaces, one of which plane surfaces extends parallel and adjacent to a side wall of'said troughand another of which plane surfaces extends at a finite angle withrespect to said first mentioned plane surface, at least two of which plane surfaces have slopes which differ with respect to each other and the projections of all of said plane surfaces leading towards a common line,'and electrode means within said trough mounted for slidable movement alongsaid side wall adjacent to said first mentioned plane surface.

11. In an analogue representing a subterranean system having axial symmetry and comprising a well containing a material having; a physical property afiecting flow of a medium in said subterranean system and subterranean strata having,different values of a physical property affecting the flow of said medium in said subterranean system, the combination: which comprises a trough having side walls and a bottom, said bottom comprising a plurality of plane surfaces, one of which plane surfaces extends parallel and adjacent to a side wall of said trough and another of which plane surfaces'extends at a finite angle with respect to said first mentioned plane surface, at least two? of which plane surfaces have slopes which differ with respect to. each other and the projections of all ofsaid planesurfaces leading towards a common line within said, trough, a unitary body of solution of electrolyte filling; said trough to the level of said common line, and electrode meanswithin said trough mounted for slidable: movement; along said side wall adjacent to said first mentioned plane surface and-contacting said unitary body-of solution of electrolyte;

Referenceszflitedzin the file. of this patent UNITED STATES PATENTS Wolf et al Oct. 2, 1951 UNITED STATES PATENT OFFICE Certificate of Correction Patent No. 2,824,689

Julius S. Aronofsky It is hereby certified that error appears in the printed specification of the above numbered patent requiring correction and that the said Letters Patent should read as corrected below.

Column 4:, 1ine 1l, for uncertainity read uncertainty-; line 43, for area 76 read area 7 0-; llnes 62 to 64, Equation 3, should appear as shown below instead of as in the patenta( be I b be 6x 0w) by 6y 0 line 67, for b is the equivalent read a is the equivalent; column 5, line 10, for the effective conductivity 5 read the effective conductivity 0-; column 11, line 45, for heat passage read heat passing.

Signed and sealed this 4th day of November 1958.

Attest KARL H. AXLINE,

Attesting Oficer. Uommisszoner of Patents.

February 25,1958 I ROBERT c. WATSON, 

